My company makes physical rehabilitation equipment based on various pendulums, fulcrums, levers that swing, push, rotate, etc.. To date the machines have been purely mechanical devices. We are now looking to add some intelligence to these machines, e.g:

Each machine would have 1-3 sensors that we need to take the readings.

We then need to perform trigonometric calculations for torque, maximum velocity, etc. (probably on the processor on the machine or even in the cloud (?)

then..

- send the data (raw or processed) to the cloud server which will analyze the data
- receive feedback from the cloud server based on analysis of the data received from the machine
- have a touch screen to display:
a) the progression of the program (reppetition count, maybe a torque curve, etc) and combine it with b) data received from the server (e.g adjustments to the program)

it also need to:

- authenticate users on the system using something like a RFID band or maybe a fingerprint scanner

And possibly:

- have a camera for recording of exercise progression and stream the video to the cloud - not sure about that though...

What platform(s) or combinations would you recommend to use to manage the cost of the hardware in under few hundred dollars that can perform something like that?

Looking now at Arduino, Raspberry Pi, but unsure if these can do it best...

There are examples on the Web using image processing with the BeagleBone Black ( also dubbed "BBB" ), which is a core with an architecture few more powerful if compared to the Raspberry PI, but it costs almost 10$ more expensive.